February 17
Russell Miller,
CUNY
Computability and the Absolute Galois Group of $\mathbb Q$
Fix a computable presentation $\overline{\mathbb Q}$ of the algebraic closure of the rational numbers. The absolute Galois group of the rational numbers, which is precisely the automorphism group of the field $\overline{\mathbb Q}$, may then be viewed as a collection of paths through a finite-branching tree. Each individual automorphism has a Turing degree. We will use known results in computability to try to build natural countable elementary subgroups of the absolute Galois group. Several intriguing questions in number theory will appear as we measure the extent to which we succeed in doing so.